For the case of a 2x2 matrix, it's probably easiest to achieve your objective by adding a spacer directive such as [1ex] after the first line-break instruction.
Incidentally, the use of $$ to initiate and terminate display-math mode in LaTeX is heavily deprecated. See Why is \[ ... \] preferable to $$ ... $$ for a longer discussion of this issue.

\documentclass{article}
\usepackage{amsmath} % for 'bmatrix' environment and '\text' macro
\begin{document}
\[
\begin{bmatrix}
1-\frac{1-\sigma_2}{2} & \frac{1-\sigma_2}{2} \\
\frac{1-\sigma_2}{2} & 1-\frac{1-\sigma_2}{2}
\end{bmatrix}
\quad\text{versus}\quad
\begin{bmatrix}
1-\frac{1-\sigma_2}{2} & \frac{1-\sigma_2}{2} \\[1ex] % <-- observe "[1ex]" spacer
\frac{1-\sigma_2}{2} & 1-\frac{1-\sigma_2}{2}
\end{bmatrix}
\]
\end{document}